with \((x,y) \in \mathbb {R}^k \times \mathbb {R}^{n-k}\)  as a special case, since (REF ) can be written in the form (REF ) after setting \(z = (x,y)^\top \) , \(f(z) = \tilde{f}(x,y,0)\) , \(N(z) = (O_{k,n-k}, I_{n-k})^\textnormal {T}\)  and \(G(z,\varepsilon ) = (\tilde{g}(x,y,\varepsilon ), \tilde{f}(x,y,\varepsilon ) - \tilde{f}(x,y,0))^\textnormal {T}\) . Conceptually, the formulation presented in [1]} can be considered as a coordinate-independent extension of earlier formulations for standard form systems (REF ) in e.g. [2]}, [3]}, [4]}, motivated by a large number of applications for which fast-slow dynamics arises in systems that are `beyond the standard form', see e.g. [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}.
